The astronomical unit (AU, equal to the mean radius of the Earth’s orbit) is 1.4960E11 m, and a year is 3.1557E7 s. Newton’s gravitational constant is G=6.6743E-11 m^3kg^-1s^-2. Calculate the mass of the Sun in kilograms. (Recalling or looking up the mass of the Sun does not constitute a solution of this problem.)
To calculate the mass of the Sun using the given information, we can use Newton's law of universal gravitation:
F = (G * mSun * mEarth) / r^2
Where:
F is the gravitational force between the Sun and Earth,
G is the gravitational constant,
mSun is the mass of the Sun,
mEarth is the mass of the Earth (which we can approximate to be around 5.972 × 10^24 kg),
and r is the distance between the Sun and Earth, which is equal to one astronomical unit (AU).
Since Earth's gravitational force towards the Sun is given by:
F = mEarth * a
Where a is the acceleration due to the Sun's gravity, we can equate these two equations:
mEarth * a = (G * mSun * mEarth) / r^2
Simplifying the equation, we get:
a = (G * mSun) / r^2
Now, we can rearrange the equation to solve for the mass of the Sun, mSun:
mSun = (a * r^2) / G
Given that a = 9.81 m/s^2 (approximate acceleration due to gravity on Earth), and r = 1.4960 × 10^11 m (AU in meters), and G = 6.6743 × 10^-11 m^3kg^-1s^-2, we can substitute these values into the equation:
mSun = (9.81 * (1.4960 × 10^11)^2) / (6.6743 × 10^-11)
Evaluating the expression, we get:
mSun ≈ 1.989 × 10^30 kg
Therefore, the mass of the Sun is approximately 1.989 × 10^30 kilograms.